Skip to main content

Path-family build plan

CalculusPath

CalculusPath is a rigorous but learner-facing path from limits and derivatives to multivariable calculus, vector calculus, series, ODEs, and the real-analysis on-ramp needed by TheoremPath, ProbabilityPath, StatisticsPath, ActuaryPath, and CUDAPath.

Family graph JSONConcept registry JSON

Launch target

Topic pages
45
Diagnostics
1
Exercises
500

Canonical reference stack

  • Apostol, Calculus, Volumes 1-2.
  • Spivak, Calculus.
  • Stewart, Calculus: Early Transcendentals.
  • Thomas, Weir, Hass, Thomas' Calculus.
  • Marsden and Tromba, Vector Calculus.
  • Hubbard and Hubbard, Vector Calculus, Linear Algebra, and Differential Forms.
  • Abbott, Understanding Analysis.
  • Rudin, Principles of Mathematical Analysis.
  • Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems.
  • Paul's Online Notes as an open applied reference, never as the only authority.

Modules

Layer 0

Functions, Limits, and Notation

Establish the language of functions, limits, continuity, and rates before differentiation.

  • Functions and graphs
  • Limit laws
  • One-sided limits
  • Continuity
  • Epsilon-delta proofs
  • Asymptotic notation for calculus

Layer 1

Derivatives and Local Linearization

Make derivatives feel like local linear models, not symbol games.

  • Derivative definition
  • Product, quotient, and chain rules
  • Implicit differentiation
  • Inverse functions
  • Taylor approximation
  • Optimization in one variable

Layer 2

Integration and Accumulation

Connect antiderivatives, area, change of variables, and accumulation models.

  • Riemann sums
  • Fundamental theorem of calculus
  • Substitution
  • Integration by parts
  • Improper integrals
  • Numerical integration

Layer 3

Sequences, Series, and Approximation

Teach convergence and approximation with enough rigor for analysis and probability.

  • Sequences and monotone convergence
  • Series convergence tests
  • Power series
  • Taylor series with remainder
  • Uniform convergence intuition

Layer 4

Multivariable Calculus

Prepare learners for optimization, statistics, ML, and differential geometry.

  • Partial derivatives
  • Gradient and directional derivative
  • Jacobian and Hessian
  • Multivariable chain rule
  • Constrained optimization and Lagrange multipliers
  • Change of variables

Layer 5

Vector Calculus and ODE On-Ramp

Bridge calculus to geometry, physics, dynamical systems, and ML models.

  • Line integrals
  • Divergence and curl
  • Green's theorem
  • Stokes' theorem
  • Divergence theorem
  • First-order ODEs

First topics

Layer 1 / tier 1

The Chain Rule

Diagram: Input -> inner function -> outer function -> rate propagation

Differentiate nested elementary functions and explain the inner-rate factor.

Layer 4 / tier 1

Gradient and Directional Derivatives

Diagram: Level curves with gradient normal to the curve

Compute steepest-ascent direction and compare with coordinate partials.

Layer 3 / tier 1

Taylor Series with Remainder

Diagram: Function approximated by local polynomial plus error band

Approximate exponential and bound the remainder.

Layer 2 / tier 1

Integration and Change of Variables

Diagram: Domain interval remapped through substitution

Use substitution to compute probability-density integrals.

Layer 4 / tier 2

Lagrange Multipliers

Diagram: Objective contours tangent to a constraint curve

Solve a constrained maximum and identify regularity assumptions.

Linking and pedagogy

  • CalculusPath owns computational and learner-facing calculus; TheoremPath owns rigorous theorem statements and ML-theory uses.
  • The first visible cross-links should target chain rule, gradients, Hessians, Taylor expansion, integration/change of variables, and vector calculus for backpropagation and optimization.
  • ActuaryPath links to interest, annuities, survival integrals, and hazard-rate calculus; StatisticsPath links to likelihood derivatives and Fisher information.
  • GeometryPath receives vector-calculus and differential-forms handoffs only when the geometric structure is genuinely load-bearing.
  • Every topic starts with a worked computational example, then gives the formal statement, then a diagnostic that maps common errors to prerequisite gaps.
  • Question pools are generated as original problems; AP/CLEP/IB/A-Level questions are cited only by source and never reproduced.
  • Adaptive practice should use mastery bands for derivative rules, integration methods, multivariable notation, and convergence tests before moving to mixed problems.